AP Stats Practice FRQ: Mixed Units #4

A researcher in a city with a large public train system wondered if the rent prices of one-bedroom apartments was related to the distance from the nearest train station. From a list of 250 similarly-sized one-bedroom apartments in the city, the researcher selected a simple random sample of 20 apartments. The researcher then measured the walking distance, in minutes, to the nearest train station and created a scatterplot comparing the walking distances to the advertised weekly rent, in dollars. A scatterplot and the output from computer regression software are shown below.
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a. Explain a procedure by which the researcher may have selected the simple random sample.

b. Describe the association between walking distance from the nearest train station and weekly rent for the apartments included in the sample.

c. Interpret the value of the coefficient of determination (r-squared) in the context of this problem.

d. Before examining the data, a second researcher makes a prediction that for each additional minute of walking distance, the weekly rent will decrease by approximately $2. A 95% confidence interval for the slope of the regression line is constructed from the data, and is found to be (-2.471, -1.845). Does the confidence interval contradict the researcher’s claim? Justify your answer.

e. The second researcher wants to conduct a similar study to the first researcher. However, in the second researcher’s study, a mixture of one, two, and three-bedroom apartments were selected. Do you expect the value of r-squared for the second study to be greater than, less than, or equal to the value of r-squared in the first study? Justify your response.

a. Give each apartment a unique number from 1-250. Randomly generate 20 unique nonrepeating numbers and measure the walking distance in minutes to the nearest train station, as well as take note of the weekly rent.
b. The association between walking distance from the nearest train station and weekly rent for the apartments is a strong negative linear association. As the distance from the nearest train station increases by 1 minute, the weekly rent tends to decrease by $2.158.
c. 92.1% of the variation in weekly rent can be attributed to the variance in distance from the nearest train station.
d. This confidence interval does not contradict the researcher’s claim since -2 is included in the interval, it is a plausible value for the slope of the regression line, and -2 is what the second researcher predicted.
e. I would expect the r-squared for this second study to be less than the r-squared of the first study because there is now going to be more variation in the data since it includes more different types of apartments. The variation in weekly rent is going to be less attributed to the distance from the nearest train station because the association will be weaker.

a. The researcher could have assigned each apartment in the sample a number from 1-250 and used a random number generator to choose 20 apartments, taking out any repeats.

b. There is a strong, negative linear correlation between he weekly rent and distance from the nearest train station. (should I say correlation or association?)

c. 92.1% of the variability of in the weekly rent dollars can be accounted for by the variability in the distance from nearest train station.

d. No it does not, because $2 falls within the given confidence interval.

e. I expect the r-squared value to be less than the first study. This is because r will decrease since there will be lower residual because there will be an increased variability in the weekly rent since it can differ depending on the number of bedrooms. If r decreases, r squared decreases as well.

Good work again - you’re ready for today!

Little things:
in (a), you must specify that the 20 numbers should be contained within the interval of 1-250 (you’re leaving yourself open to other numbers the way you phrase it).

In (b), either word is ok - correlation or association. I tend to use the language that is included in the question to play it safe.

c, d, and e give fully appropriate explanations!

a. The researcher could have numbered the 250 similarly-sized apartments from 1-250 and wrote the numbers on an equally sized piece of paper. Then, the researcher could put these slips in a hat and she it very well. Then the researcher can pick 20 slips of paper which would represent the 20 apartments that will be used in the sample.
b. The association between walking distances from the nearest train station and weekly rent for the apartments included in the sample is strong, negative, and linear with no suspected outliers. It is strong as the correlation coefficient= square root of r^2 = square root of .921 is .9596 which shows us that the data points follow a strong pattern and linear pattern. We know the data points have a negative association as the slope of the linear regression line is -2.158.
c. The coefficient of determination, says that 92.1% of the variation in weekly rent is accounted for by the regression line when x= distance from nearest train station.
d. Since the confidence interval contains -2, the researchers claim is not contradicted.
e. We expect the r-squared to be less than the first study. This is because we will have more variability in the weekly rent since there are more types of apartments which vary in weekly rent. This leads us to have less of an association between the weekly rent and the train station.

Nicely done!

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